Local homology
The homology groups (cf. Homology group)
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defined at points , where
is homology with compact support. These groups coincide with the direct limits
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over open neighbourhoods of
, and for homologically locally connected
they also coincide with the inverse limits
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The homological dimension of a finite-dimensional metrizable locally compact space over
(cf. Homological dimension of a space) coincides with the largest value of
for which
, and the set of such points
has dimension
.
Let be the differential sheaf over
defined by associating with each open set
the chain complex
. The groups
are the fibres of the derived sheaves
. For generalized manifolds,
for
. In this case the homology sequence of the pair
with coefficients in
coincides with the cohomology of the pair
with coefficients in the sheaf
(Poincaré–Lefschetz duality). The similar facts for the local cohomology of locally compact spaces do not hold.
References
[1] | E.G. Sklyarenko, "On the theory of generalized manifolds" Math. USSR Izv. , 5 : 4 (1971) pp. 845–858 Izv. Akad. Nauk SSSR Ser. Mat. , 35 (1971) pp. 831–843 |
[2] | A.E. [A.E. Kharlap] Harlap, "Local homology and cohomology, homology dimension and generalized manifolds" Math. USSR Sb. , 25 : 3 (1975) pp. 323–349 Mat. Sb. , 96 (1975) pp. 347–373 |
Comments
References
[a1] | A. Dold, "Lectures on algebraic topology" , Springer (1972) pp. Sect. IV.3 |
Local homology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_homology&oldid=47681